EM 388F Fracture Mechanics, Spring 2008
Rui Huang
8. Nonlinear Fracture Mechanics I: J-integral and Crack-tip Field
References:
J. W. Hutchinson, Notes on Nonlinear Fracture Mechanics (http://imechanica.org/node/755);
Alan Zehnder, Lecture Notes on Fracture Mechanics (http://hdl.handle.net/1813/3075).
Deformation theory of plasticity. Under a monotonic loading, elastic-plastic
deformation of a material may be described by a nonlinear stress-strain relationship within the
context of small strains, which is essentially a nonlinear elasticity theory sometimes called
hypoelasticity. Despite its limitations, the deformation theory provides useful insights into the
elastic-plastic stress field near a crack tip and initiation of crack growth in ductile materials (low
strength and high toughness) under monotonic loads.
The nonlinear stress-strain relationship can be developed by assuming a strain energy
density function, w(ε ij ) , such that
σ ij =
∂w
∂ε ij
Under uniaxial tension, a power-law stress-strain relationship is often adopted for the
convenience of analysis:
ε ⎛σ ⎞
=⎜ ⎟
ε 0 ⎜⎝ σ 0 ⎟⎠
n
where σ 0 may be considered as an effective tensile yield stress and ε 0 the associated tensile
strain. The power exponent n is typically greater than 1. For many metals, n = 10~25. For the
limiting cases, the material is perfectly plastic (no hardening) when n = ∞ and is perfectly elastic
when n = 1 .
Under multi-axial stresses, the power-law relation takes the general form:
ε ij 3 ⎛ σ e ⎞
= ⎜ ⎟
ε 0 2 ⎜⎝ σ 0 ⎟⎠
n −1
sij
σ0
3
1
where sij = σ ij − σ kk is the deviatoric stress tensor and σ e =
sij sij is the effective shear
2
3
stress. The material is assumed to be incompressible under the deformation theory of plasticity.
Energy release rate. Consider a cracked specimen. Under
monotonic loading, the strain energy is
Δ
U (a, Δ ) = ∫ P (u )du
0
where P(u) is a nonlinear function depending on the crack length a.
The potential energy is
Π(a, P ) = U − PΔ
Similar to LEFM, the energy release rate per unit thickness
for straight-ahead crack growth can be defined as
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EM 388F Fracture Mechanics, Spring 2008
Rui Huang
⎛ ∂U ⎞
G = −⎜
⎟
⎝ ∂a ⎠ Δ
⎛ ∂Π ⎞
Or equivalently, G = −⎜
⎟ . Two points are noted for the definition of energy release rate:
⎝ ∂a ⎠ P
⎛ ∂U ⎞
In LEFM, Π = −U , and thus G = ⎜
(i)
⎟ . The same is not true in nonlinear fracture
⎝ ∂a ⎠ P
mechanics.
(ii)
Strictly speaking, the definition of the energy release rate is only meaningful for
linear or nonlinear elastic materials, for which the strain energy in the body can be
fully released upon unloading. For elastic-plastic materials, part of the strain energy is
unrecoverable due to plasticity. As the crack advances, only the elastic part of the
strain energy is released. Nevertheless, a critical energy release rate may be
determined as the toughness for the initiation of crack growth under monotonic
loading.
J-integral. The contour integral previously discussed under the condition of small-scale
yielding for LEFM can be extended to nonlinear fracture mechanics under the condition that
there exists a strain energy density function in form of w(ε ij ) . In a 2D specimen (plane stress or
plane strain), the J-integral
⎛
∂u ⎞
J = ∫ ⎜⎜ wn1 − ti i ⎟⎟ds
C
∂x1 ⎠
⎝
is path-independent as long as the contour starts on one crack face and ends on the other. It was
further shown that J = G (Budiansky and Rice, J. Appl. Mech., pp. 201, 1973).
Crack-tip field (HRR solution). Consider a semi-infinite crack in an infinite body. Two
sets of dimensional analyses lead to a formal solution for the stress field near the crack tip. First,
given a single loading parameter P, the stress scales linearly with the applied load, i.e., σ ij ~ P .
By the power-law relationship, the strain scales with the load as ε ij ~ P n . Integration of the strain
tensor with respect to the spatial coordinates gives the displacement with the same scaling,
ui ~ P n . Consequently, the strain energy density, w = ∫ σ ij dε ij ~ P n +1 , and the energy release rate
or J-integral, J ~ wa . The dimensional consideration thus leads to a general form of the energy
release rate (Ilyushin theorem):
n +1
⎛σ ⎞
J = Ωaσ 0ε 0 ⎜⎜ appl ⎟⎟
⎝ σ0 ⎠
where Ω is a dimensionless parameter that depends on the specimen geometry. This is
σ 2a
. Numerical
analogous to the dimensional form of the energy release rate in LEFM: G = Ω
E
values of Ω for a number of crack specimens are available in the ASTM handbook by C. F. Shih
(1974).
Next, take the contour C of the J-integral to be a circle of radius r centered at the crack
tip, and thus
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EM 388F Fracture Mechanics, Spring 2008
Rui Huang
π ⎛
∂u ⎞
J = ∫ ⎜⎜ wn1 − σ ij n j i ⎟⎟rdθ
−π
∂x1 ⎠
⎝
For J to be independent of the path radius r and to have a finite value as r → 0 , it is necessary
that
∂u
f (θ )
as r → 0
wn1 − σ ij n j i =
∂x1
r
1
Therefore, the strain energy density scales inversely with r, i.e., w ~ . With the power-law
r
relationship, the stress and strain must have the following scaling as r → 0 :
ε ij ~ r
−
n
n +1
−
1
, σ ij ~ r n +1
Both the strain and stress are singular, but the singularity for n > 1 differs from the square-root
singularity in LEFM for which n = 1. On the other hand, integration of the strain with respect to r
1
gives the displacement with a scaling: ui ~ r n +1 , which is non-singular.
Combining the two sets of dimensional analyses, we obtain the following:
σ ij ~ P r
−
1
n +1
, ε ij ~ P n r
−
n
n +1
1
, ui ~ P n r n +1
Since J ~ P n +1 independent of r, we replace the loading parameter P in the above scaling with J
as P ~ J
1
n +1
, and obtain that
n
1
n
⎛ J ⎞ n +1
⎛ J ⎞ n +1
σ ij ~ ⎜ ⎟ , ε ij ~ ⎜ ⎟ , ui ~ J n +1 r n +1
⎝r⎠
⎝r⎠
1
The crack-tip field can be written as
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⎛ J ⎞ n +1 ~
⎟⎟ σ ij (θ , n )
σ ij = σ 0 ⎜⎜
⎝ σ 0ε 0 r ⎠
⎛
n
n +1
⎞ ~
⎟⎟ ε ij (θ , n )
σ
ε
r
⎝ 0 0 ⎠
where the circumferential distributions are given by the dimensionless functions σ~ij (θ , n ) and
ε~ (θ , n ) . The above crack-tip solution as well as numerical results for the circumferential
ε ij = ε 0 ⎜⎜
J
ij
functions were obtained independently by Hutchinson (J. Mech. Phys. Solids 16, pp. 13-31,
1968) and Rice and Rosengren (J. Mech. Phys. Solids 16, pp. 1-12, 1968), thus called HRR
solution. It was noted that the dimensionless functions depend on whether plane stress or plane
strain condition holds in the plastic zone. Similar to LEFM, the θ-dependence of the HRR field
can be obtained separately for mode I and mode II cracks, and the analogous field for mode III
can also be obtained analytically.
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EM 388F Fracture Mechanics, Spring 2008
Rui Huang
Remarks: (i) The HRR solution is a nonlinear extension of the K-field solution in
LEFM. Here, J plays a similar role as K measuring the intensity of the singular stress field at the
crack tip that depends on external loading and specimen geometry. The small scale yielding
condition in LEFM is now replaced by a small-scale process zone condition. (ii) The
dimensionless functions σ~ij (θ , n ) and ε~ij (θ , n ) are unique for finite n (strain hardening). For
n → ∞ (perfectly plasticity), however, they are not unique but depend on the full solution to the
boundary value problem. This is a consequence of the fact that the field equations of perfectly
plasticity are hyperbolic while those for finite n are elliptic. (iii) The HRR solution is valid only
in an annular region around the crack tip, where the small strain deformation theory of plasticity
is applicable. When this region is large compared to the fracture process zone and the region of
large strains governed by finite strain plasticity, the crack tip field can be sufficiently
characterized by a single parameter J. This restriction strongly depends on the crack
configuration, strain hardening exponent, and plane strain or plane stress conditions.
Deeply cracked test specimens. These specimens are designed such that a simple
formula for J can be used to determine J as a function of load-displacement data directly from
the measurements. For example, for the deeply
cracked bend specimen as shown in the figure
with c/b < 0.5, the J-integral was shown to be
given by (Rice et al., 1973)
2 θ
M (θ cr )dθ cr
c ∫0
where M is the applied bending moment per unit thickness of the specimen and θ cr = θ − θ 0 : θ is
J=
the total angle of rotation at the load point, θ 0 is the reference rotation for an uncracked
specimen (a = 0), and θ cr is the contribution due to the presence of the crack. Therefore, the Jintegral is simply 2/c times the area under the bending moment M vs θ cr curve that can be
measured experimentally.
A similar expression for J-integral applies to deeply
cracked compact tension specimen:
2 Δ
J = ∫ P (Δ cr )dΔ cr
c 0
Where P is the applied load per unit thickness of the specimen
and c = b − a is the length of the uncracked ligament. Often
Δ cr is replaced by the total displacement Δ since the
reference displacement Δ 0 for an uncracked compact tension
specimen is usually negligible.
The bending and compact tension specimens can be
used to determine the critical J for initiation of crack growth as well as the resistance curve J vs
Δa for small amount of crack growth (Shih et al., 1978).
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